Trackpad works but can't click.

In aggregate, this might indicate a hardware problem; possibly a swelling battery. A battery that’s plumping can prevent the trackpad from registering, or always registering as being clicked.

As for using it as is, the external keyboard and mouse is the usual, and—if that’s not working, as indicated here—then you will want to make an appointment with Apple for a look at the hardware.

Primary and secondary click can be reset in system preferences / system settings, yes.

That's how users can select which mouse buttons are used for which hand.

As can tap and two-fingered tap to click, with trackpads and such.

This does seem either a damaged or stuck trackpad, either itself or maybe due to the battery swelling. If the trackpad is stuck down, it's stuck clicking. I've also met folks that "misplaced" Bluetooth mice around their desk, piled some papers onto those, the buttons got triggered, and that was indeed "fun" to troubleshoot.

Sight down the sides of and edges of the case, and see if any distortion is visible, too.

I'd try a wired keyboard, and NVRAM and SMC resets, too.

Control the pointer using Mouse Keys on Mac - Apple Support